Quantum field theory on curved Spacetime - Ahmed YOUSSEF
No one knows how to write a full quantum gravity theory. â» But we expect the ... external field. â» The coupling to an external field furnish energy that can.
Cosmological particle creation Field equation Particle creation
YOUSSEF Ahmed Director : J. Iliopoulos LPTENS Paris Laboratoire de physique th´ eorique de l’´ ecole normale sup´ erieure
QFT on de Sitter spacetime Classical de Sitter spacetime QFT on de Sitter spacetime
Final notes
Why QFT on curved spacetime Introduction
I In its usual formulation QFT simply ignores gravity
Why and when ? Consequences Example
Cosmological particle creation
I No one knows how to write a full quantum gravity theory
Field equation Particle creation
QFT on de Sitter spacetime
I But we expect the existence a semi-classical regime where
one can only quantize matter fields and keep gravity classical (cf a theory of quantum matter interacting with a classical electromagnetic field) I ψmatter coupled to classical gµν ⇐⇒ ψmatter lives on curved
spacetime
Classical de Sitter spacetime QFT on de Sitter spacetime
Final notes
Consequences of the coupling to an external field I The coupling to an external field furnish energy that can
create particles : Schwinger effet in QED coupled to an ~ field external E m2 Pe+ e− pair creation ∝ exp − eE
I More importantly, the notion of particle is ambiguous.
Remember that origin of the particle concept in QFT is an asymptotic one I I
Free QFT −→ E = space of stationnary solutions E has a Fock space structure =⇒ particle interpretation of theory
Introduction Why and when ? Consequences Example
Cosmological particle creation Field equation Particle creation
QFT on de Sitter spacetime Classical de Sitter spacetime QFT on de Sitter spacetime
Final notes
Consequences of the coupling to an external field I The same reasoning doesn’t hold in an interacting theory
Introduction Why and when ? Consequences Example
Cosmological particle creation Field equation Particle creation
QFT on de Sitter spacetime Classical de Sitter spacetime QFT on de Sitter spacetime
Final notes
I Example of QCD I I
:
Free theory −→ Quarks Interacting theory −→ hadrons
I Particle notion defined asymptotically in a free theory
Particle notion defined asymptotically in a flat spacetime
Quantum driven harmonic oscillator I equation of motion
q¨ + ω 2 q = J(t) q˙ = p˙ =
p −ω 2 q + J(t) r i ω q(t) ∓ p(t) a± = and a− (t = 0) = a− in 2 ω
Introduction Why and when ? Consequences Example
Cosmological particle creation Field equation Particle creation
QFT on de Sitter spacetime Classical de Sitter spacetime QFT on de Sitter spacetime
Final notes
I Solution
i −iωt a− (t) = a− +√ in e 2ω
t
Z 0
dτ J(τ )eiω(τ −t)
Quantum driven harmonic oscillator I 2 asymptotic regions
Introduction
−
a (t) =
− a− out = ain + J0
I 2 vacuum
−iωt a− in e − aout e−iωt
i J0 = √ 2ω
if t ≤ 0 if t ≥ T T
Z
dτ J(τ )eiωτ
=0 = J0 |0in i
|0in i 6= |0out i N (t) = a+ (t)a− (t)
I Particle (exitation) creation
h0in | N (t) |0in i =
Field equation Particle creation
Classical de Sitter spacetime QFT on de Sitter spacetime
a− out |0out i a− out |0in i
=0
Cosmological particle creation
QFT on de Sitter spacetime
0
|0in i and |0out i
a− in |0in i
Why and when ? Consequences Example
0 |J0 |2
if t ≤ 0 if t ≥ T
Final notes
Real scalar field I Minimal coupling
Z S=
Introduction
√ 1 d4 x −g g µν ∂µ φ∂ν φ − m2 φ2 2
+ m2 φ = 0
√ 1 φ = √ ∂µ g µν −g∂ν φ −g
Why and when ? Consequences Example
Cosmological particle creation Field equation Particle creation
QFT on de Sitter spacetime Classical de Sitter spacetime QFT on de Sitter spacetime
I Mode decomposition I
Minkowski : Poincar´e invariance gives a privileged coordinate system (t, x, y, z) ( P φ(t, ~x) = ~k a~k u~k (t, ~ x) + a~†k u~∗k (t, ~ x) u~k
I
~
∝ e−ik.~x e−iωt
Curved spacetime : many different mode decomposition X X φ(x) = ai ui (x) + a†i u∗i (x) = a ¯i u ¯i (x) + a ¯†i u ¯∗i (x) i
i
Final notes
Bogoliubov transformation I 2 different vacuum
Introduction
ai |0i a ¯i |¯ 0i
=0 =0
Why and when ? Consequences Example
∀i ∀i
Cosmological particle creation Field equation Particle creation
I {ui } and {¯ ui } complete bases of states
u ¯j =
X
QFT on de Sitter spacetime
αji ui + βji u∗i
Classical de Sitter spacetime QFT on de Sitter spacetime
i
( =⇒
I ai |¯ 0i =
P
j
ai a ¯j
P ∗ † = j αji a ¯j + βji a ¯j P ∗ ∗ † = i αji ai − βji aj
∗ ¯ βji |1j i 6= 0 if a βji 6= 0
I Created particle number
Ni = a†i ai
h¯ 0| Ni |¯ 0i =
X j
|βji |2
Final notes
Particle creation in spacially flat FRW
Why and when ? Consequences Example
I Conformally flat spacetime
ds2 = dt2 −a2 (t)dx2
dη = 2
gµν = a (η)ηµν
dt ds2 = a2 (η) dη 2 − dx2 a(t) √ −g = a4 (η)
I field equation ~ 1 uk (η, ~ x) = √ eik.~x χk (η) 2π
χ ¨k + k2 + a2 (η)m2 χk = 0
I Exact soution in terms of hypergeometric functions if
a2 (η) = A + B tanh(ρ η)
Introduction
Cosmological particle creation Field equation Particle creation
QFT on de Sitter spacetime Classical de Sitter spacetime QFT on de Sitter spacetime
Final notes
Particle creation in spacially flat FRW
Why and when ? Consequences Example
I We impose Minkowskian modes as η → ±∞
(
uin k uout k
= · · · −→ = · · · −→
√ 1 ei(kx−ωin η) 4πωin 1 √ ei(kx−ωout η) 4πωout
as as
Introduction
η → −∞ η →∞
Cosmological particle creation Field equation Particle creation
QFT on de Sitter spacetime
I One can then find analytically αk and βk such that out uin + βk uout∗ k = αk uk −k
I Number of created particles
N=
X k
|βk |2
Classical de Sitter spacetime QFT on de Sitter spacetime
Final notes
Classical de Sitter spacetime I de Sitter spacetime : maximally symmetric spacetime with
isotropic and homogeneous spacial sections, positive scalar curvature Example : Flat spatial sections 2
2
ds = −dt + e I dSd may be realized in M
−X02
+
d,1
X12
2Ht
2
d~ x
+ ··· +
Here the O(d, 1) symmetry is manifest
=l
Why and when ? Consequences Example
Cosmological particle creation Field equation Particle creation
as the hyperboloid Xd2
Introduction
2
QFT on de Sitter spacetime Classical de Sitter spacetime QFT on de Sitter spacetime
Final notes
2 point function I Free field theory, so all the information is in the 2 point
G(X, Y ) = h0| φ(X)φ(Y ) |0i
Introduction Why and when ? Consequences Example
function. For instance the Wightman function ∆dSd − m2 G = 0
Cosmological particle creation Field equation Particle creation
I G(X, Y ) = G (P (X, Y )) where P (X, Y ) is the de Sitter
invariant length. Hypergeometric equation : ¨ + d − zd G˙ − m2 G = 0 with z(1 − z)G 2
z=
1+P 2
I Solutions : a one parameter family of de Sitter invariant
Green functions Gα corresponding to a one parameter family of de Sitter invariant vacuum states |αi Gα (X, Y ) = hα| φ(X)φ(Y ) |αi
QFT on de Sitter spacetime Classical de Sitter spacetime QFT on de Sitter spacetime
Final notes
Thermal radiation I A geodesic observer x(τ ) equipped with a detector of
Hamiltonian and energy eigenstates H |Ej i = Ej |Ej i
Introduction Why and when ? Consequences Example
Cosmological particle creation Field equation Particle creation
I The geodesic observer measures a thermal bath of particles
when the field φ is in the vacuum state |0i : the field-detector coupling induces a thermally populated energy levels Ni ∝ e−βEi
I The de Sitter temperature is
T =
1 2πl
QFT on de Sitter spacetime Classical de Sitter spacetime QFT on de Sitter spacetime
Final notes
Final notes I One can compute the vectorial and spinorial propagator too
Introduction
I Instead of a matter field we can consider linearized gravity
Perturbative quantum gravity is non renormalizable. This is a short distance property that is independent of the large scale shape of spacetime
I
But one can still treat it as an effective field theory and get the first quantum corrections
I
The graviton hµν propagator on de Sitter has an infrared pathology even at the tree level
QFT on de Sitter spacetime Classical de Sitter spacetime QFT on de Sitter spacetime
Final notes
Final notes I One can try to consider the back reaction of quantum fields
(matter and gravitons) on spacetime Gµν = 8πG hTµν i I Cosmological constant problem : gravity couples to any
form of energy. So a naive renormalization of the vacuum energy is not possible and Z ΛPlanck E0 1p 2 = d3 k k + m2 ≈ Λ4Planck ≈ 1094 g.cm3 ! V 2
Introduction Why and when ? Consequences Example
Cosmological particle creation Field equation Particle creation
QFT on de Sitter spacetime Classical de Sitter spacetime QFT on de Sitter spacetime
Final notes
I The Stone von Neumann theorem breaks down in infinite
dimensional context (field theory). Infinitley many inequivalent representations of the quantum algebra exists and no Poincar´e invariance to pick one −→ algebraic approach to QFT on curved space time
Final notes Introduction
I Probably profound link between gravity, the quantum and
thermodynamics I dS/CFT correspondence I Finally one more reason to think that QFT is the quantum
theory of fields and not a quantum theory of particles. To mention also : Rovelli’s global/local particles in QFT
Why and when ? Consequences Example
Cosmological particle creation Field equation Particle creation
QFT on de Sitter spacetime Classical de Sitter spacetime QFT on de Sitter spacetime
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